Efficient Bootstrap Tests for the Goodness of Fit in Covariance Structure Analysis

Asymptotic chi-square tests, such as the normal theory likelihood ratio test, are often used to evaluate the goodness-of-fit of a covariance structure analysis model. Another approach is to use the bootstrap test, which is known to have the desired asymptotic level if model restrictions are taken into account in designing a resampling algorithm. The bootstrap test is, however, computationally very tedious and the problem of nonconvergence and improper solutions often arise in bootstrap resampling. In this paper, we propose a bootstrap test which is based on an approximation, by a quadratic form, to the minimum value of a discrepancy function calculated from each bootstrap sample. Hence, the proposed bootstrap test is efficient in the sense of the amount of computing needed and is free from the problem of nonconvergence and improper solutions with resampling. A Monte Carlo experiment is conducted to compare the performance of the proposed method with that of asymptotic chi-square tests for each combination of three distributions and four sample sizes.